# Determining Concavity

> AP Calculus AB · AP Calculus AB CED Unit 5
> Source: https://www.owlsprep.com/study/ap-calculus-ab-u5-determining-concavity/

This guide aligns to AP Calculus AB expectations, covering formal concavity definitions, the second derivative rule, inflection point identification, and finding concavity from first derivative graphs.

**Prerequisites:** Computing first and second derivatives of differentiable functions; Interpreting and writing open interval notation; Testing the sign of a function on an interval

## Learning objectives

- Define concavity formally and informally for differentiable functions
- Apply the second derivative rule to find intervals of concavity
- Correctly identify inflection points by checking for concavity changes
- Determine concavity and inflection points from a graph of the first derivative
- Avoid common exam pitfalls related to concavity and inflection points

## Core Definition of Concavity

Concavity describes the direction of curvature of a differentiable function $f(x)$ over an interval, and it is a core high-frequency skill on the AP Calculus AB exam. Unit 5 (Analytical Applications of Differentiation) makes up 10–15% of total exam weight, and concavity questions appear in both multiple-choice and free-response sections.

Informally, concave up intervals are shaped like a cup ($\cup$) that "holds water", and concave down intervals are shaped like a cap ($\cap$) that "spills water".

**Concavity** — Formal definition based on tangent line position: a function is concave up on an open interval if all tangent lines on the interval lie below the function, and concave down if all tangent lines lie above the function.

*Example:* $f(x) = x^2$ is concave up across its entire domain.

> **note**
>
> Determining concavity is a prerequisite for identifying inflection points and using the second derivative test for local extrema, two other heavily tested Unit 5 skills.

## Second Derivative Rule & Interval Testing

Concavity describes how the slope of the tangent line (the first derivative $f'(x)$) changes as $x$ increases. If a function is concave up, its slope increases as $x$ increases; if it is concave down, its slope decreases as $x$ increases. Because the second derivative $f''(x)$ measures the rate of change of $f'(x)$, this gives a direct testable relationship between the sign of $f''$ and concavity.

**Second Derivative Rule for Concavity** — If $f$ is twice differentiable on an open interval $I$:
1. If $f''(x) > 0$ for all $x \in I$, then $f$ is concave up on $I$
2. If $f''(x) < 0$ for all $x \in I$, then $f$ is concave down on $I$

1. Compute the second derivative $f''(x)$
2. Find all candidate split points where $f''(x) = 0$ or $f''(x)$ is undefined (these split the domain into open test intervals)
3. Test the sign of $f''(x)$ in each interval
4. Assign concavity based on the sign of $f''(x)$

**Worked example:** Find all intervals where $f(x) = x^3 - 6x^2 + 9x + 2$ is concave up and concave down.

1. Compute the first derivative:

   $$f'(x) = 3x^2 - 12x + 9$$
2. Compute the second derivative:

   $$f''(x) = 6x - 12 = 6(x-2)$$
3. Find candidate split points: $f''(x)$ is defined for all real $x$, and $f''(x)=0$ only at $x=2$, splitting the domain into two intervals: $(-\infty, 2)$ and $(2, \infty)$
4. Test $x<2$ with $x=0$:

   $$f''(0) = -12 < 0$$
5. Therefore, $f$ is concave down on $(-\infty, 2)$
6. Test $x>2$ with $x=3$:

   $$f''(3) = 6 > 0$$
7. Therefore, $f$ is concave up on $(2, \infty)$

> **tip**
>
> On FRQ questions, you must explicitly reference the sign of the second derivative in your justification (e.g., "concave up on $(2, \infty)$ because $f''(x) > 0$ for all $x$ in this interval") to earn full credit.

## Identifying Inflection Points

An inflection point is a point on the graph of $f$ where concavity changes from up to down or down to up. For an inflection point to exist at $x=c$, two non-negotiable conditions must be met: (1) $f(c)$ is defined (the point $(c, f(c))$ lies on the graph of $f$), and (2) the sign of $f''(x)$ (and thus concavity) changes across $x=c$.

A common student misconception is that inflection points only occur where $f''(c)=0$, and that all points with $f''(c)=0$ are inflection points. This is incorrect: inflection points can also occur where $f''(c)$ is undefined (as long as $f(c)$ exists and concavity changes), and $f''(c)=0$ does not guarantee a concavity change.

**Worked example:** Find all inflection points of $f(x) = x^4 - 4x^3 + 6x^2$.

1. Compute first and second derivatives:

   $$f'(x) = 4x^3 - 12x^2 + 12x \\ f''(x) = 12x^2 - 24x + 12 = 12(x-1)^2$$
2. Generate candidate points: $f''(x)$ is defined for all real $x$, and $f''(x)=0$ only at $x=1$, so the only candidate is $x=1$
3. Test for concavity change at $x<1$ with $x=0$:

   $$f''(0) = 12 > 0$$
4. Test for concavity change at $x>1$ with $x=2$:

   $$f''(2) = 12 > 0$$
5. Concavity does not change at $x=1$, so there are no inflection points for this function

> **tip**
>
> AP exam questions almost always include a distractor that relies on the mistake of assuming all $f''(c)=0$ points are inflection points. Always confirm the sign change.

## Concavity from a Graph of $f'(x)$

The AP exam frequently tests the skill of determining concavity when given only the graph of the first derivative $f'(x)$, not an explicit formula for $f(x)$. We use the core relationship: $f''(x)$ is equal to the slope of the tangent line to the graph of $f'(x)$ at $x$.

This gives us a simple rule: if $f'(x)$ is increasing on an interval, its slope is positive, so $f''(x) > 0$, so $f(x)$ is concave up. If $f'(x)$ is decreasing on an interval, its slope is negative, so $f''(x) < 0$, so $f(x)$ is concave down. Inflection points on $f(x)$ correspond exactly to local extrema (peaks or valleys) on the graph of $f'(x)$.

**Worked example:** The graph of $f'(x)$, the first derivative of $f(x)$, is a parabola opening downward with vertex at $x=3$, crossing the x-axis at $x=1$ and $x=5$. What interval is $f(x)$ concave up on?

1. Recall: $f(x)$ is concave up when $f'(x)$ is increasing, because the slope of $f'$ equals $f''(x)$
2. A downward-opening parabola increases to the left of its vertex and decreases to the right of its vertex
3. The vertex of this parabola is at $x=3$, so $f'(x)$ is increasing on $(-\infty, 3)$ and decreasing on $(3, \infty)$
4. Therefore, $f(x)$ is concave up on $(-\infty, 3)$

> **warning**
>
> Do not confuse the sign of $f'(x)$ (which tells you if $f(x)$ is increasing or decreasing) with the slope of $f'(x)$ (which tells you concavity of $f(x)$). Ignore the vertical position of $f'$ when finding concavity.

**Check your understanding**

Test your understanding:

1. The second derivative of a function $f$ is given by $f''(x) = (x+1)(x-3)^2$. For what values of $x$ does the graph of $f$ have an inflection point?

   - A) $x=-1$ only
   - B) $x=3$ only
   - C) $x=-1$ and $x=3$
   - D) No inflection points

   *Why:* Correct: $f''$ changes sign from negative to positive only at $x=-1$. It stays positive on both sides of $x=3$, so no concavity change occurs there.

## Common pitfalls

- **Wrong:** Stating $x=c$ is an inflection point only because $f''(c)=0$, without checking that concavity changes at $x=c$
  - Why it fails: Students memorize that inflection points occur where $f''=0$, so they stop there and skip the required sign check
  - Correct: After finding all candidate points, always test the sign of $f''$ on both sides of the candidate to confirm a sign change, and confirm $f$ is defined at $c$
- **Wrong:** Confusing the sign of $f'(x)$ with the slope of $f'(x)$ when finding concavity from an $f'$ graph, e.g., saying $f$ is concave up when $f'(x)>0$
  - Why it fails: Students mix up the two uses of the first derivative, one for monotonicity of $f$ and one for concavity of $f$ via the slope of $f'$
  - Correct: When asked for concavity from an $f'$ graph, only pay attention to whether $f'$ is increasing or decreasing, not whether it is positive or negative
- **Wrong:** Using closed intervals (e.g., $(-\infty, 2]$) for intervals of concavity on FRQ answers
  - Why it fails: Students assume endpoints should be included because $f$ is defined there, but concavity is defined for open intervals
  - Correct: Always write intervals of concavity as open intervals on the AP exam, unless explicitly asked otherwise
- **Wrong:** Ignoring candidate inflection points where $f''(c)$ is undefined
  - Why it fails: Students only look for where $f''(x)=0$ and forget that points where the second derivative does not exist can still be inflection points
  - Correct: When finding candidates, always include all points in the domain of $f$ where $f''(x)=0$ or $f''(x)$ is undefined, then test each for a concavity change
- **Wrong:** Claiming a decreasing function cannot be concave up, e.g., saying $f(x)=e^{-x}$ is concave down because it is decreasing
  - Why it fails: Students incorrectly associate 'decreasing' with 'concave down', confusing slope direction with curvature direction
  - Correct: Always separate monotonicity (from $f'$ sign) and concavity (from $f''$ sign): any combination of increasing/decreasing and concave up/down is possible

## Cheatsheet

| Category | Rule / Formula | Notes |
| --- | --- | --- |
| Concave Up (Open Interval $I$) | $f''(x) > 0$ for all $x \in I$, or $f'(x)$ increasing on $I$ | Curves upward (cup shape $\cup$), tangent lines lie below $f$ |
| Concave Down (Open Interval $I$) | $f''(x) < 0$ for all $x \in I$, or $f'(x)$ decreasing on $I$ | Curves downward (cap shape $\cap$), tangent lines lie above $f$ |
| Inflection Point Conditions | 1. $f(c)$ is defined; 2. $f''$ changes sign at $x=c$ | Candidates are $f''(c)=0$ *or* $f''(c)$ undefined; $f''(c)=0$ alone is not sufficient |
| Concavity from $f'(x)$ Graph | $f$ concave up when $f'$ increasing; $f$ concave down when $f'$ decreasing | Ignore the sign of $f'$ for concavity; sign of $f'$ tells you if $f$ is increasing/decreasing |
| Inflection Points from $f'(x)$ Graph | Inflection points on $f$ occur at local extrema of $f'$ | Local maxima/minima of $f'$ are where slope of $f'$ changes sign |
| Second Derivative Test for Extrema | If $f'(c)=0$: $f''(c)<0 \implies$ local max; $f''(c)>0 \implies$ local min | Test is inconclusive if $f''(c)=0$ or undefined; use first derivative test in that case |
| Intervals of Concavity | Always use open intervals | Concavity is defined for open neighborhoods around each point, so endpoints are excluded |

## What's next

Now that you can determine concavity and identify inflection points, the next immediate step is applying this knowledge to the second derivative test for local extrema, which relies entirely on concavity to classify critical points. This topic is also the foundation for full curve sketching, where you combine information about monotonicity, concavity, and inflection points to draw accurate graphs of functions from derivative information. Across Unit 5, all analytical applications of differentiation build on concavity to connect derivative behavior to global function properties, which is tested heavily on the exam. Without mastering how to correctly determine concavity, you will not be able to earn full credit on graph analysis FRQ questions.

- [Second Derivative Test](https://www.owlsprep.com/study/ap-calculus-ab-u5-second-derivative-test/)
- [Sketching graphs of f, f', f''](https://www.owlsprep.com/study/ap-calculus-ab-u5-sketching-graphs-of-f-f/)
- [Connecting f, f', f'' qualitatively](https://www.owlsprep.com/study/ap-calculus-ab-u5-connecting-f-f-f-qualitatively/)

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